Well, when a player with less than a certain number of credited NFL games gets released/waived/cut (all mean the same thing), then he must clear waivers before he is a UFA, free to sign with any team. I'm not 100% sure the number of games this is, but its usually first/second and possibly third year players. The waiver priority for this particular cut day (as it has been throughout the offseason) is based on last years final standings. Thus, we get first claim on every player on waivers. If we claim a player, he becomes a Dolphin. Now, if the Rams, who have the second priority, claim a player, there are two options. If we put in a claim, then we get him, since we're above them, but if we dont, then they are awarded that player.
This doesnt apply to a guy like, for instance, Rosey Colvin, who is immediately unrestricted when released.
Hope this helps.